A problem that appeared on AMM

Prove that a square cannot be dissected into an odd number of triangles of equal area.



ELMO 2012 SL N9

Are there positive integers m,n such that there exist at least 2012 positive integers x such that both m-x^2 and n-x^2 are perfect squares?

Putnam 2014 B6

Let f:[0,1]\to\mathbb{R} be a function for which there exists a constant K>0 such that |f(x)-f(y)|\le K|x-y| for all x,y\in [0,1]. Suppose also that for each rational number r\in [0,1], there exist integers a and b such that f(r)=a+br. Prove that there exist finitely many intervals I_1,\dots,I_n such that f is a linear function on each I_i and [0,1]=\bigcup_{i=1}^nI_i.